If `veca, vecb,vecc` are three non-coplanar vectors such that `veca xx vecb=vecc,vecb xx vecc=veca,vecc xx veca=vecb`, then the value of `|veca|+|vecb|+|vecc|` is

Options

A. `1//3`
B. 1
C. 3
D. 6

Explanation

To solve the problem, we start with the given conditions involving the vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\):

1. \(\vec{a} \times \vec{b} = \vec{c}\)
2. \(\vec{b} \times \vec{c} = \vec{a}\)
3. \(\vec{c} \times \vec{a} = \vec{b}\)

We need to find the value of \(|\vec{a}| + |\vec{b}| + |\vec{c}|\).

### Step 1: Use the properties of the cross product

From the first equation, we can take the magnitude of both sides:

\[ |\vec{a} \times \vec{b}| = |\vec{c}| \]

Using the property of the cross product, we know that:

\[ |\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \sin \theta \]

where \(\theta\) is the angle between \(\vec{a}\) and \(\vec{b}\). Thus, we have:

\[ |\vec{a}| |\vec{b}| \sin \theta = |\vec{c}| \]

### Step 2: Apply the same approach to the other equations

For the second equation, we take the magnitude:

\[ |\vec{b} \times \vec{c}| = |\vec{a}| \]

This gives us:

\[ |\vec{b}| |\vec{c}| \sin \phi = |\vec{a}| \]

where \(\phi\) is the angle between \(\vec{b}\) and \(\vec{c}\).

For the third equation, we also take the magnitude:

\[ |\vec{c} \times \vec{a}| = |\vec{b}| \]

This gives us:

\[ |\vec{c}| |\vec{a}| \sin \psi = |\vec{b}| \]

where \(\psi\) is the angle between \(\vec{c}\) and \(\vec{a}\).

### Step 3: Establish relationships between the magnitudes

From the equations derived, we have:

1. \(|\vec{a}| |\vec{b}| \sin \theta = |\vec{c}|\)
2. \(|\vec{b}| |\vec{c}| \sin \phi = |\vec{a}|\)
3. \(|\vec{c}| |\vec{a}| \sin \psi = |\vec{b}|\)

### Step 4: Assume magnitudes are equal

Let’s assume \(|\vec{a}| = |\vec{b}| = |\vec{c}| = k\) for some positive scalar \(k\). Then substituting this into the equations gives:

1. \(k^2 \sin \theta = k\) ⇒ \(\sin \theta = \frac{1}{k}\)
2. \(k^2 \sin \phi = k\) ⇒ \(\sin \phi = \frac{1}{k}\)
3. \(k^2 \sin \psi = k\) ⇒ \(\sin \psi = \frac{1}{k}\)

### Step 5: Determine the value of \(k\)

Since \(\sin \theta\), \(\sin \phi\), and \(\sin \psi\) must all be less than or equal to 1, we have:

\[ \frac{1}{k} \leq 1 \implies k \geq 1 \]

### Step 6: Calculate the total magnitude

Now, substituting \(k = 1\):

\[ |\vec{a}| + |\vec{b}| + |\vec{c}| = k + k + k = 3k = 3 \]

Thus, the value of \(|\vec{a}| + |\vec{b}| + |\vec{c}|\) is:

\[ \boxed{3} \]